A solution of the biharmonic dirichlet problem by means of hypercomplex analytic functions

2017

Lobachevskii J Math

A commutative algebra B over the field of complex numbers with the bases {e 1 , e 2 } satisfying the conditions (e 2 1 + e 2 2 ) 2 = 0, e 2 1 + e 2 2 = 0, is considered. The algebra B is associated with the biharmonic equation. Consider a Schwartz-type boundary value problem on finding a monogenic function of the type Φ(xe 1 + ye 2 ) = U 1 (x, y) e 1 + U 2 (x, y) ie 1 + U 3 (x, y) e 2 + U 4 (x, y) ie 2 , (x, y) ∈ D, when values of two components U 1 , U 4 are given on the boundary of a domain D lying in the Cartesian plane xOy. We develop a method of its solving which is based on expressions of monogenic functions via corresponding analytic functions of the complex variable. For a half-plane and for a disk, solutions are obtained in explicit forms by means of Schwartz-type integrals. Monogenic functions in a biharmonic algebraDefinition 1. An associative commutative two-dimensional algebra B with the unit 1 over the field of complex numbers C is called biharmonic (see [1,2]) if in B there exists a basis {e 1 , e 2 } satisfying the conditions (e 2 1 + e 2 2 ) 2 = 0, e 2 1 + e 2 2 = 0 . Such a basis {e 1 , e 2 } is also called biharmonic.In the paper [2] I. P. Mel'nichenko proved that there exists the unique biharmonic algebra B, and he constructed all biharmonic bases in B. Note that the algebra B is isomorphic to four-dimensional over the field of real numbers R algebras considered by A. Douglis [3] and L. Sobrero [4].2010 Mathematical Subject Classification. 30G35; 31A30.

“…Note that hypercomplex methods and suitable "analytic" functions are used for investigation of elliptic partial differential equations in the papers [3,4,16,17,18,19,20,21,22,23].…”

Section: (K-m)-problem For Monogenic Functionsmentioning

confidence: 99%

A Schwartz-type boundary value problem in a biharmonic plane

2017

Lobachevskii J Math

“…Hypercomplex methods for investigating the biharmonic equation were developed in the papers [17,18,19,20,21,22] also.…”

Section: Introductionmentioning

confidence: 99%

Monogenic functions in the biharmonic boundary value problem

Math Methods in App Sciences

2015

We consider a commutative algebra $\mathbb{B}$ over the field of complex numbers with a basis $\{e_1,e_2\}$ satisfying the conditions $(e_1^2+e_2^2)^2=0$, $e_1^2+e_2^2\ne 0$. Let $D$ be a bounded domain in the Cartesian plane $xOy$ and $D_{\zeta}=\{xe_1+ye_2 : (x,y)\in D\}$. Components of every monogenic function $\Phi(xe_1+ye_2)=U_{1}(x,y)\,e_1+U_{2}(x,y)\,ie_1+ U_{3}(x,y)\,e_2+U_{4}(x,y)\,ie_2$ having the classic derivative in $D_{\zeta}$ are biharmonic functions in $D$, i.e. $\Delta^{2}U_{j}(x,y)=0$ for $j=1,2,3,4$. We consider a Schwarz-type boundary value problem for monogenic functions in a simply connected domain $D_{\zeta}$. This problem is associated with the following biharmonic problem: to find a biharmonic function $V(x,y)$ in the domain $D$ when boundary values of its partial derivatives $\partial V/\partial x$, $\partial V/\partial y$ are given on the boundary $\partial D$. Using a hypercomplex analog of the Cauchy type integral, we reduce the mentioned Schwarz-type boundary value problem to a system of integral equations on the real axes and establish sufficient conditions under which this system has the Fredholm property.Comment: 30 page

“…Note that in the papers [4,[19][20][21] for investigations of biharmonic functions there are other approaches, which involve commutative finite-dimensional algebras over the field R and suitable "analytic" hypercomplex functions.…”

Section: Biharmonic Monogenic Functions and Schwarz-type Boundary Valmentioning

confidence: 99%

Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations

2017

Open Mathematics

: We consider a commutative algebra B over the field of complex numbers with a basis fe 1 ; e 2 g satisfying the conditions .e (1-4)-problem for monogenic B-valued functionsˆ.xe 1 C ye 2 / D U 1 .x; y/ e 1 C U 2 .x; y/ i e 1 C U 3 .x; y/ e 2 C U 4 .x; y/ i e 2 having the classic derivative in the domain D D fxe 1 C ye 2 W .x; y/ 2 Dg: to find a monogenic in D functionˆ, which is continuously extended to the boundary @D , when values of two component-functions U 1 , U 4 are given on the boundary @D. Using a hypercomplex analog of the Cauchy type integral, we reduce the (1-4)-problem to a system of integral equations on the real axes. We establish sufficient conditions under which this system has the Fredholm property and the unique solution. We prove that a displacements-type boundary value problem of 2-D isotropic elasticity theory is reduced to (1-4)-problem with appropriate boundary conditions. Biharmonic monogenic functions and Schwarz-type boundary value problems for themAn associative commutative two-dimensional algebra B with the unit 1 over the field of complex numbers C is called biharmonic (see [1,2]) if in B there exists a basis fe 1 ; e 2 g satisfying the conditions.e Such a basis fe 1 ; e 2 g is also called biharmonic. In the paper [2] I. P. Mel'nichenko proved that there exists the unique biharmonic algebra B, and he constructed all biharmonic bases in B. Note that the algebra B is isomorphic to four-dimensional over the field of real numbers R algebras considered by A. Douglis [3] and L. Sobrero [4].In what follows, we consider a biharmonic basis fe 1 ; e 2 g with the following multiplication table (see [1]):2 D e 1 C 2i e 2 ; where i is the imaginary complex unit. We consider also a basis f1; g (see [2]), where a nilpotent element D 2e 1 C 2i e 2 satisfies the equality 2 D 0 .